So they give us a function, f of x.

So let me just draw a little bit of a graph.

So x is equal to r3.

So let's see if we can make some headway on this.

That's because I've fit my parameters to the test set.

Polynom

polynom

Now let's consider the model selection problem.

Let's start with the training error.

Sometimes there's more than one, right?

lambda, such as if lambda were equal to 0.

So the easiest way to approximate it is to say, well, the simplest polynomial is just a constant, right?

Now let's plot the following figure.

One thing I could do is

Consider the following graph where the horizontal axis graphs complexity of the solution.

All right, so to do the last little bit of this proof to show that 3 colorability is NP hard, we're going to show that if we had the ability to solve 3 colorability problems in polynomial time, then we could solve three SAT problems in polynomial time as well, and so what we need to be able to do to show that is if you walk up to me with any 3 CNF formula,

For example, instead of sending a straight

So that's the area and it's a monomial.

Let's do another one like this.

Let me draw my axes.

And what we did in this with the repeated factor is true if we went to a higher degree term.

So this checks out.

What can I do with this?

No need to feed me or recharge me.

And we showed it in the beginning.

Let's keep going.

I have something shifted here.

In this case, in all of the examples we'll do, it'll be x.

And let's say, for the sake of this example, that I end up picking theta 5, the fifth order polynomial, because that has the Noah's cross validation error.

So, you should you choose a linear function, a quadratic function, a cubic function, all the way up to a 10th power polynomial?

In this video, I want to focus on a few more techniques for factoring polynomials.

It could be that NP is equal to x. It could also that P is equal to NP.

We know the exponential function.

And the reason is, what we've done is, we've fit this extra the parameter d, that is this degree of polynomial, and we'll fit that parameter d using the test set.

So in that case, I'm going to pick this fourth order polynomial model and finally what this means is that that parameter d, remember d was the degree of polynomial, right d equals 2, d equals 3, up to d equals 10.

So 5000 features seems like a lot, if you were to include the cubic, or third order known of each others, the x1, x2, x3.

It could be the case that the class NP is actually equal to the class EXP.

Okay, so you have a sense that these are considered to be pretty good algorithms.

Oh. We're not good enough?

In this video I want to do a bunch of examples of factoring a second degree polynomial, which is often called a quadratic.

So if we viewed a squared as kind of the independent variable or the x term, so now this kind of has the shape of polynomials that hopefully you're used to factoring a

So for this example, let's say for the sake of argument that it was my fourth order polynomial that had the lowest cross validation error.

whereas we're here on the right of the horizontal axis, I have much larger values of these of a much higher degree polynomial, and so here that is going to correspond to fitting much more complex functions to your training set.

This is true.

So let's try to factor each of them. So first, let's try the numerator. And I'll actually do it up here.